Related papers: A proof of the Gap Labeling Conjecture

In this paper we prove the "Tiling implies Spectral" part of Fuglede's paper for the case of three intervals. Then we prove the "Spectral implies Tiling" part of the conjecture for the case of three equal intervals as also when the…

Using recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has…

For a Gaussian measure on a separable Hilbert space with covariance operator $C$, we show that the family of conditional measures associated with conditioning on a closed subspace $S^{\perp}$ are Gaussian with covariance operator the short…

Graph polytopes arising from vertex-weighted graphs were first introduced by B\'ona, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is…

We show that the Hopf algebra of quasi-symmetric functions arises naturally as the integral Chow ring of the algebraic stack of expanded pairs originally described by J. Li, using a more combinatorial description in terms of configurations…

The Conifold Gap Conjecture asserts that the polar part of the Gromov-Witten potential of a Calabi-Yau threefold near its conifold locus has a universal expression described by the logarithm of the Barnes $G$-function. In this paper, I…

The families index theory for the overlap lattice Dirac operator is applied to derive topological features of the space of SU(N) lattice gauge fields on the 4-torus: The topological sectors, specified by the fermionic topological charge,…

By Bartle-Graves theorem every surjective map between C*-algebras has a continuous section, and Loring proved that that there exists a continuous section of norm arbitrary close to 1. Here we prove that there exists a continuous section of…

In this paper, we employ quotients of Roe algebras as index containers for elliptic differential operators to study the existence problem of Riemannian metrics with positive scalar curvature on non-compact complete Riemannian manifolds. The…

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a…

In their celebrated paper (arXiv:0906.1238), Caputo, Liggett and Richthammer proved Aldous' conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying…

The Guillemin-Sternberg conjecture states that "quantisation commutes with reduction" in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups $G$ acting on compact…

For a relative effective divisor $\mathcal{C}$ on a smooth projective family of surfaces $q:\mathcal{S}\rightarrow B$, we consider the locus in $B$ over which the fibres of $\mathcal{C}$ are $\delta$-nodal curves. We prove a conjecture by…

Harvey Friedman's gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the…

We prove that a positive-definite measure in $\mathbb{R}^n$ with uniformly discrete support and discrete closed spectrum, is representable as a finite linear combination of Dirac combs, translated and modulated. This extends our recent…

Given a densely defined and gapped symmetric operator with infinite deficiency index, it is shown how self-adjoint extensions admitting arbitrarily prescribed portions of the gap as essential spectrum are identified and constructed within a…

We provide a short and reasonably self-contained proof of Lurie's straightening equivalence, relating cartesian fibrations over a given $\infty$-category $S$ with contravariant functors from $S$ to the $\infty$-category of small…

Galois rings and orders, introduced by Futorny and Ovsienko, are embedded into fixed subrings of skew group (or monoid) rings and have many interesting applications to the structure and representation theory of algebras. The paper focuses…

We introduce a symmetric operad $\square p$ ("box-op") which describes a certain calculus of rectangular labeled ``boxes''. Algebras over $\square p$, which we call box operads, have appeared under the name of fc multicategories in work by…

We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type.…